ADMM [1]} alternatively minimizes the augmented
Lagrangian function (REF ) over two blocks of variables,
leading to an efficient operator splitting framework for nonlinear programs with
nonsmooth objective function and equality constraints. Hence
ADMM solves the original problem (REF ) when reformulated as problem
(REF ) with a sequence of easier sub-problems. Specifically, ADMM
solves problem (REF ) by alternatively minimizing (REF )
over variables \(\lbrace {\theta ^c}, {\widetilde{\theta }^d} \rbrace  \) , and
\(\lbrace  \delta , \mathbf {z} \rbrace  \) . This can be equivalently converted into
3 sub-problems over variables \(\lbrace {\theta ^c}, {\widetilde{\theta }^d}\rbrace \) ,
\(\delta \)  and \(\mathbf {z}\) , respectively. ADMM decomposes the
optimization variables into two blocks and alternatively minimizes the augmented
Lagrangian function (REF ) in the following manner at any ADMM
iteration \(t\) :

\(\left\lbrace  \theta ^c{}^{(t+1)}, \widetilde{\theta }^d{}^{(t+1)} \right\rbrace & = \operatornamewithlimits{arg\,min}_{\theta ^c, \widetilde{\theta }^d} \mathcal {L} \left(\mathbf {z}{}^{(t)}, \theta ^c, \widetilde{\theta }^d, \delta {}^{(t)},\lambda {}^{(t)}\right) \\ \left\lbrace  \delta {}^{(t+1)}, \mathbf {z}{}^{(t+1)} \right\rbrace & = \operatornamewithlimits{arg\,min}_{\delta , \mathbf {z}} \mathcal {L}\left(\mathbf {z}, \theta ^c{}^{(t+1)}, \widetilde{\theta }^d{}^{(t+1)}, \delta ,\lambda {}^{(t)}\right) \\ \lambda {}^{(t+1)} & =\lambda {}^{(t)} + \rho \left(\widetilde{\theta }^d{}^{(t+1)} - \delta {}^{(t+1)}\right).\) 
